Natural porous media, such as subsurface reservoirs containing hydrocarbons, are typically highly heterogeneous and complex geological formations. High-resolution geological models, which often are composed of millions of grid cells, are generated to capture the detail of these reservoirs. Current reservoir simulators are encumbered by the level of detail available in the fine-scale models and direct numerical simulation of subsurface fluid flow on the fine-scale is usually not practical. Various multiscale methods, which account for the full resolution of the fine-scale geological models, have therefore been developed to allow for efficient fluid flow simulation.
Multiscale methods include multiscale finite element (MSFE) methods, mixed multiscale finite element (MMSFE) methods, and multiscale finite volume (MSFV) methods. All of these multiscale methods can be applied to compute approximate solutions at reduced computational cost. While each of these methods reduce the complexity of a reservoir model by incorporating the fine-scale variation of coefficients into a coarse-scale operator, each take a fundamentally different approach to constructing the coarse-scale operator.
The multiscale finite volume (MSFV) method is based on a finite volume methodology in which the reservoir domain is partitioned into discrete sub-volumes or cells and the fluxes over the boundaries or surfaces of each cell are computed. Since the fluxes leaving a particular cell are equivalent to the fluxes entering an adjacent cell, finite volume methods are considered to be conservative. Thus, the accumulations of mass in a cell are balanced by the differences of mass influx and outflux. Accordingly, mass conservation is strictly honored by multiscale finite volume (MSFV) methods, which can be very important in some reservoir simulation applications such as when a mass conservative fine-scale velocity field is needed for multiphase flow and transport simulations.
The multiscale finite element (MSFE) and mixed multiscale finite element (MMSFE) methods are based on a finite element scheme, which breaks the reservoir domain into a set of mathematical spaces commonly referred to as elements. Physical phenomenon within the domain is then represented by local functions defined over each element. These methods are not mass conservative in a strict sense due to their underlying formulation, however, some finite element methods have been able to account for this shortcoming by coupling the pressure and velocity basis functions, such as in mixed multiscale finite element (MMSFE) methods. However, such methods are computationally expensive and typically are not practical for use in commercial reservoir simulators.